Argh.
Sorry.  I started reading this before enough, or any, coffee was consumed.

Anthony - that is some wicked cool stuff right there.
Thanks for that - I really enjoyed watching that guy bang out a nicely aligned
tile set in no time at all.   Very educational!

Mike - I think the two concepts are separate, but related.

The diamond-square algorithm can get you a nice terrain:

http://www.paulboxley.com/blog/2011/03/terrain-generation-mark-one
http://www.playfuljs.com/realistic-terrain-in-130-lines/
http://www.javaworld.com/article/2076745/learn-java/3d-graphic-java--render-fractal-landscapes.html

And it won't tile _if you generate it completely randomly_ .

So maybe what you do is take a cue from the video showing how it's done in
Photoshop and use that to keep the opposite edges of the large square the same.
Whatever's at 0,0 is what ought to be at 128,0.   Whatever's at 32,0 is what
32,128 ought to be.   That way if you did that double-half-displacement thing,
the edges would be identical.
You could probably do that at the time of generation or maybe even as a
post-generation interpolation across a wider swath.

I think if you check out the guy's tile set with the dirt and grass, you'll see
that it's a radial pattern / texture - akin to a circle in a square, but not
quite circumscribed.   The edges always match, and you get a nice solid center
tile.

If I'm following Anthony with his idea of generating shades of grey - then maybe
he's suggesting using a texture map.  Make one greyscale tile set, and then use
POV-Ray to put the right terrain on the different greys to generate the
transitions.   (I admit, I haven't thought it out completely, but I think that
might be what he's suggesting)

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